Find the percentage decrease in the weight of the body when taken to a depth of `32 km` below the surface of earth. Radius of the earth is `6400 km`.

Options

A. 0.6%
B. 0.2%
C. 0.5%
D. 0.7%

Explanation

To find the percentage decrease in the weight of a body when taken to a depth of 32 km below the surface of the Earth, we can follow these steps:

### Step 1: Understand the relationship between weight and gravity
Weight (W) of a body is given by the formula:
\[ W = mg \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.

### Step 2: Define the change in weight
When the body is taken to a depth of \( d = 32 \) km, the new weight \( W' \) can be expressed as:
\[ W' = mg' \]
where \( g' \) is the acceleration due to gravity at that depth.

### Step 3: Determine the formula for gravity at a depth
The acceleration due to gravity at a depth \( d \) below the surface of the Earth is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where \( R \) is the radius of the Earth.

### Step 4: Substitute the values
Given:
- \( d = 32 \) km
- \( R = 6400 \) km

Substituting these values into the formula:
\[ g' = g \left(1 - \frac{32}{6400}\right) \]

### Step 5: Simplify the expression
Calculating the fraction:
\[ \frac{32}{6400} = 0.005 \]
Thus,
\[ g' = g \left(1 - 0.005\right) = g \times 0.995 \]

### Step 6: Calculate the percentage change in weight
The percentage change in weight can be expressed as:
\[ \text{Percentage change} = \frac{W - W'}{W} \times 100 \]
Substituting for \( W' \):
\[ \text{Percentage change} = \frac{mg - mg'}{mg} \times 100 = \frac{g - g'}{g} \times 100 \]

### Step 7: Substitute \( g' \) into the percentage change formula
Using \( g' = g \times 0.995 \):
\[ \text{Percentage change} = \frac{g - (g \times 0.995)}{g} \times 100 \]
\[ = \frac{g(1 - 0.995)}{g} \times 100 \]
\[ = 0.005 \times 100 = 0.5\% \]

### Conclusion
The percentage decrease in the weight of the body when taken to a depth of 32 km below the surface of the Earth is **0.5%**.

---